






'''simple docstring'''





import unittest

import numpy as np



def        __snake_case						(__UpperCAmelCase       ,	__UpperCAmelCase       ,	__UpperCAmelCase       ,	__UpperCAmelCase = None       ,	):





	"""simple docstring"""
	lowerCamelCase_					:  int						=    np.shape(__UpperCAmelCase    )
	lowerCamelCase_					:  List[str]						=    np.shape(__UpperCAmelCase    )
	lowerCamelCase_					:  int						=    np.shape(__UpperCAmelCase    )

	if shape_a[0] != shape_b[0]:
		lowerCamelCase_					:  Any						=    (
		    '''Expected the same number of rows for A and B. '''
		    F"""Instead found A of size {shape_a} and B of size {shape_b}"""
		)
		raise ValueError(__UpperCAmelCase    )

	if shape_b[1] != shape_c[1]:
		lowerCamelCase_					:  List[Any]						=    (
		    '''Expected the same number of columns for B and C. '''
		    F"""Instead found B of size {shape_b} and C of size {shape_c}"""
		)
		raise ValueError(__UpperCAmelCase    )

	lowerCamelCase_					:  str						=    pseudo_inv
	if a_inv is None:
		try:
			lowerCamelCase_					:  Union[str, Any]						=    np.linalg.inv(__UpperCAmelCase    )
		except np.linalg.LinAlgError:
			raise ValueError(
			    '''Input matrix A is not invertible. Cannot compute Schur complement.'''    )

	return mat_c - mat_b.T @ a_inv @ mat_b



class   lowerCAmelCase__ ( unittest.TestCase  ):

			def 						__UpperCamelCase				(					self					:		str	) ->  None:







				"""simple docstring"""






				lowerCamelCase_					:  str						=    np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]	)
				lowerCamelCase_					:  List[str]						=    np.array([[0, 3], [3, 0], [2, 3]]	)
				lowerCamelCase_					:  List[Any]						=    np.array([[2, 1], [6, 3]]	)

				lowerCamelCase_					:  Optional[int]						=    schur_complement(UpperCamelCase_     ,		UpperCamelCase_     ,		UpperCamelCase_	)

				lowerCamelCase_					:  List[str]						=    np.block([[a, b], [b.T, c]]	)

				lowerCamelCase_					:  Dict						=    np.linalg.det(UpperCamelCase_	)
				lowerCamelCase_					:  Optional[int]						=    np.linalg.det(UpperCamelCase_	)
				lowerCamelCase_					:  Union[str, Any]						=    np.linalg.det(UpperCamelCase_	)

				self.assertAlmostEqual(UpperCamelCase_     ,		det_a * det_s	)

			def 						__UpperCamelCase				(					self					:		Dict	) ->  None:







				"""simple docstring"""






				lowerCamelCase_					:  Tuple						=    np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]	)
				lowerCamelCase_					:  List[str]						=    np.array([[0, 3], [3, 0], [2, 3]]	)
				lowerCamelCase_					:  str						=    np.array([[2, 1], [6, 3]]	)

				with self.assertRaises(UpperCamelCase_	):
					schur_complement(UpperCamelCase_     ,		UpperCamelCase_     ,		UpperCamelCase_	)





			def 						__UpperCamelCase				(					self					:		str	) ->  None:







				"""simple docstring"""






				lowerCamelCase_					:  Optional[Any]						=    np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]	)
				lowerCamelCase_					:  Any						=    np.array([[0, 3], [3, 0], [2, 3]]	)
				lowerCamelCase_					:  int						=    np.array([[2, 1, 3], [6, 3, 5]]	)

				with self.assertRaises(UpperCamelCase_	):
					schur_complement(UpperCamelCase_     ,		UpperCamelCase_     ,		UpperCamelCase_	)


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			unittest.main()












'''simple docstring'''







def 						_A      ( UpperCAmelCase = 1      ,UpperCAmelCase = 1000						):






   '''simple docstring'''







   A__       =     1
   A__       =     0

   for divide_by_number in range(UpperCAmelCase      ,digit + 1						):
      A__       =     []
      A__       =     numerator
      for _ in range(1      ,digit + 1						):
         if now_divide in has_been_divided:
            if longest_list_length < len(UpperCAmelCase						):
               A__       =     len(UpperCAmelCase						)
               A__       =     divide_by_number
         else:
            has_been_divided.append(UpperCAmelCase						)
            A__       =     now_divide * 10 % divide_by_number

   return the_digit


# Tests
if __name__ == "__main__":
    import doctest

    doctest.testmod()


